A water change looks difficult once your plants have been growing for a while. Those roots become a web of chaos after a while but a complete water change every month or so is strongly advised and should be done. Granted this will not work for everyone but I have found this to be the easiest way with a AeroGarden 6 without the need to purchase a siphon and without causing any ill or trying effects to the plants.
Step 1: Fill the sink about half way with room temperature water
Step 2: Carefully remove the cover from the water basin. If your plants have been around for a while the roots will have likely become meshed together with one another. The trick here is to remove the support bracket that goes across the basin so you do not have to untangle the roots from one another.
Step 3: Pour the water from the basin into the sink which should fill it up nearly the rest of the way
Step 4: Rest the tray with the plants on top of the sink
Step 5: Clean out the basin (I use the bath tub) and if needed the airstone as well. The airstone can have buildup on it so be sure to check it for any obstructions. Once cleaned refill it all the way making very sure the water is room temperature. If it is too cold or hot you can easily put your plants into shock which will likely not kill them but will definitely make them unhappy for a couple days.
You can see where I had to remove the left black support bar in this picture (when I removed the plants)
Step 6: Add the nutrients, lift the tray with the plants from the sink and place it onto the basin. As you place it back onto the basin be sure to place the black support (that you may have removed earlier) through the opening of the roots and back into the slots in the basin.
That is it. Water replaced without a mess and no need for anything outside of your bathroom.